From what I understand electrons aren't necessarily a 'physical entity', rather they're better described as an area of higher probability of interaction (i.e., orbitals). Relativistic quantum chemistry, however, is based upon the observation that electrons, especially those in higher orbitals, are moving at some appreciable fraction of the speed of light.

Help me reconcile the concept of an orbital with an electron's velocity. Should this velocity be thought of more as a frequency?

Electrons are particles and 'physical entities' as far as anything is. In fact, they're elementary particles, so more particle-ish than most particles. What I mean by 'elementary particle' here is that they have no known constituent parts, no observed internal structure. Electrons interact with things as if they were an infinitesimal point in space.

But they're very lightweight, so they behave very quantum-mechanically. That means they don't have a well-defined position in space; they're 'spread out' and we can only say the probability of where they're likely to be is. Heisenberg's Uncertainty Principle prohibits things from having an exact position and/or momentum - not just electrons, but everything. All particles, even macroscopic objects. It's just that the uncertainty is so small relative the mass and size of large objects, that it's undetectable, but at the atomic/molecular scale the spreading out of electrons is large enough to be quite important. But in QM they still interact as if they were point charges, just point charges that have a number of possible locations. If you want to calculate the electrical force from it, you'd calculate the force from a point charge at location x and multiply that with the probability of the electron being at location x, and sum that up over all the possible locations.

An orbital, on the other hand, is not an electron. It's a description of the energy states that electrons can have when they're bound in an atom or molecule, and together with that, the probabilities of where the electron in that orbital is likely to be when it's in that state. Electrons are fundamentally indistinguishable (unless they have opposite 'spin'), so while you can talk about orbitals being 'occupied' or not, you fundamentally can't say which electron is occupying it, which is why the orbitals are what's interesting.

As with their position, and for the same reason, they don't have an exact momentum either; it's 'spread out'. The average momentum of an electron in an atom is actually zero (which makes sense, as if they had a net momentum in some direction, they should go flying off). But the average magnitude of the momentum (p2) is not zero. Their average kinetic energy (which is Ek = p2/2m, just as in classical mechanics) is thus not zero.

But in Special Relativity, the kinetic energy in terms of momentum is Ek = sqrt(p2c2+m2c4) - mc2 , for which p2/2m is just a first-order approximation. So that's the relationship you have to use instead, and the difference between classical and quantum mechanics is that the momentum p is a simple variable classically, while in quantum mechanics, you have to deal with a statistical probability-distribution over the possible values of p.

They 'look' quite different. Free, moving electrons have a net momentum for starters, and their probability distribution (which is thus moving through space) can have an almost-arbitrary shape. Except that the shape of the probability distribution (regardless of situation) is related to the kinetic energy (again per the uncertainty principle), the more precisely located a particle is to a specific region of space, the more the momentum is spread out, and vice-versa.

There are a couple of ways we can look imagine the velocity of a quantum particle.

One way is to look at the expectation value of the momentum of the electron. This looks at the spread of momentum values and picks out the average value. We can then use the classical momentum formula to find a velocity.

Another way is to simply treat the orbits in a classical or semi-classical way (like the bohr model).

A good approximation can come from taking simple path integrals of an approximate orbit of the electron and finding the velocity from there (semi-classical approach).

Ultimately, I don't think you are correct in saying that electrons aren't physical entities. They are, just not classical ones. They can very much have a velocity even if it isn't a set classical value and picture.